- Email: [email protected]
Height limits in top down and bottom up wave environments
COASTAL ENGINEERING ELSEVlER
Coastal Engineering
32 (1997)
Technical
247-254
Note
Height limits in top down and bottom up wave environments Ray Nelson School qf Cil’il Engineering,
*
lJniuer& of New South Wales, Unirxr-sity Collrgr, Academy, Canberra, ACT 2600, Australia
Amtmliarr
llpfcrlce Fnrcc
Keceived 23 January 19Y7; accepted 20 June 1997
Abstract This paper isolates two environments or processes in which maximum stable wave heights can be attained in water of constant depth. In fop down environments the maximum stable wave height, compatible with the depth, is approached from above, while in bottom up bottom up wave trains are shown to conform to the same upper height limit criteria as naturally occurring top down wave trains, that limit being the same function developed from monochromatic laboratory waves. However, important differences are found in the shallow and transitional wave content of the wave populations generated by the two different processes. 0 1997 Elsevier Science B.V. Keywords: Waves;
Constant depth; Depth limited wave heights; Design wave heights
1. Introduction Design height
wave
that can
heights
in neat-shore
be sustained
regions
are often
in the prevailing
water
governed depths,
by the maximum
establishing
wave
a requirement
for depth limited, design wave height criteria. Most project locations are associated with finite bed slopes, but there will always be a significant number of sites located in constant water depth regions (horizontal bed). In Australia, for example, fixed and floating structures are often located on the flat tops of coral reef platforms. Navigation
* Tel.:
t 61-Z-62688330;
FAX:
+61-Z-62688337;
0378.383Y/97/$17,00 0 1997 Elsevier PII s0378-3839(97)00024-0
E-mail:
[email protected].
Science B.V. All rights reserved
R. Nelson/Coastal
248
Engineering
32 (1997) 247-254
aids have been constructed on offshore sand banks in water depths of up to 8 m where the bed level has varied by less than 300 mm over an area of 4 km2. A submerged ocean pipeline has been designed for a cyclone prone area in a constant water depth location so extensive that a water depth of 15 m is persistent over a 10 km length of the pipeline. Nelson, 1982, 1985, 1987, 1994 has shown that the depth limited wave heights that can be sustained in water of constant depth are much lower than those that can be sustained even on very mild slopes (say = 0.01 and even less). The subject therefore has very important design implications. This paper reviews the previous laboratory and field work done by the author on this topic. The work is then extended by identifying two natural processes that produce stable wave trains in water of constant depth, introduces new field data, and compares the depth limited nature of the waves produced by the two processes. lmportant similarities and differences are found. The two wave environments producing the two processes have been called top down and bottom up wave environments. In top down environments, the maximum stable wave height compatible with the existing constant water depth, is approached from above. For example, waves shoaled into the constant depth region, at heights not sustainable by the water depth, break, dissipate energy, and reduce in height until the maximum stable wave height is achieved. High waves propagated from deep water into the constant water depth on a coral reef platform is an example of a top clown wave environment. Bottom up waves approach the maximum stable wave height, compatible with the existing water depth, from below. The waves are at their lowest height when they first exist in the constant water depth region, and must grow in height to achieve the maximum stable wave height. Depth limited wind waves generated on a large shallow lake of uniform depth is an example of a bottom up wave environment.
2. Regular wave data in water of constant depth In a series of papers, Nelson, 1982, 1985, 1987, 1994 developed a function from laboratory data, defining the maximum possible value of H/h for stable oscillatory waves, propagating in water of constant depth. All the data were from monochromatic (regular) wave experiments. The data and the derived function are shown in Fig. 1 (CERC, 1984; Goda, 1964; Keating and Webber, 1977; Le Mehaute et al., 1968; Nelson, 1985, 1994) where use has been made of a non-linearity parameter (F,) after Swart and Loubser (1979). This parameter is a function of T, h and H as given in Eq. (1) where T is the wave period, h is the water depth, and H is the wave height. F, = ‘
g1.25H0.50
T2.50
,$I .I5
This is a particularly useful parameter because it is a function of variables that may be directly measured at a point, even over a sloping bed, a property not achievable if wave length (L) was involved (e.g., Ursell’s parameter). The main implication is that waves of equal F, have approximately the same relative wave shape. Approximate values of F,
R. Nelson/Coastal
Engineering
32 (1997) 247-254
249
corresponding to the classical classification of shallow, transitional, and deep water waves are; shallow water waves F, > 500 transitional water waves 10 < F, < 500 deep water waves F, < 10 Some of the data shown in Fig. 1 are from data sets in which the waves are known to have been at or near the limit of stability (wave breaking). These data are labelled according to the data sets from which they were extracted. For data sets where no distinction was made between waves at or near the limit of stability and those well below that limit, all the data have been plotted under the common label others. The envelope curve defining the maximum H/h ratio is described by Eq. (2). H -= h
F,
(2)
22 + 1.82F,
In shallow water, this function is asymptotic to H/h = 0.55, a value at odds with common engineering practice and the limiting H/h ratio for horizontal beds given by CERC (1984) shown in Fig. 1. This assumed a limiting H/h value of 0.78 based on the work of McGowan (1894) using solitary waves (not oscillatory waves). The heavy dotted line in Fig. 1 represents the deep water steepness limit of 0.142 (Nelson, 1994). It is known that some of the waves shown in Fig. 1 were shoaled into the region of constant water depth, while others are from experiments where a continuous horizontal floor existed between the wave generator and the wave observation station. Fig. 1 includes two data sets of limiting H/h ratios for regular waves over a bed slope of 0.01 (Goda, 1964; Nelson, 1987). They demonstrate how bed slopes as small as this influence the wave mechanics sufficiently to increase the limiting value of H/h to 0.8 and greater.
deep
/
transitiohol
shallow
co
Nelson
(1985),
slope
0.000,
data
set
1
Nelson
(1985),
slope
0.000,
data
set
2
Le
d
Mehaute
Keating
and
Others,
slope
Goda
(1964),
et
01
(1968).
Webber
(1977),
slope
Nelson
(1987),
Nelson
(1985.1994),
slope
0.010 0.010 slope
“1W
CERC
(1984),
slope
0.000
CERC
(1984).
slope
0.010
H/L=O. 142
1o4 Fc Fig. 1. Monochromatic,
laboratory
wave data.
slope
0.000
(H,‘h)max=Fc/(22+1.82Fc)
_-_-_-
slope
0.000
0.000 0.000
250
R. Nelson/Coastal
Engineering
32 (1997) 247-254
Nelson (1987) undertook laboratory experiments to determine how the limiting ratios for H/h vary on bed slopes between horizontal and 0.01. Using six slopes, it was determined that Eq. (3) described the upper limit of H/h for stable, unbroken, oscillatory waves in shallow water, where p is the bed slope angle. = 0.55 + 0.88exp(
3. Natural,
irregular,
-O.O12cot(
/3))
(3)
top down, stable waves in water of constant depth
Nelson (1994) extended the above work to the analysis of field data of wave trains propagating over a broad, horizontal, coral reef platform. 72 data sets were analyzed which covered a wide range of incident wave height and reef top water depth combinations. In about 75% of these data sets, the wave trains were shoaled by the reef front to the extent that they contained breaking waves at the start of the constant water depth region. Wave energy was then dissipated by the turbulence associated with the wave breaking process, until stable wave heights, compatible with reef top water depths, were attained. The experimental observations were made at a point where the stable, oscillatory waves had reformed. These waves are referred to as top down waues and the data will be referred to as top down duta. Wave by wave analyses of these data give rise to the results shown in Fig. 2. These take into account any effects tide or infragravity wave activity may have had in varying the effective water depth on which the wind waves were superimposed. The left side of Fig. 2 shows only one wave from each data set, namely that corresponding to the greatest value of H/h observed in each data set. The right side of Fig. 2 shows all waves from all data sets. Also shown is the envelope curve of Eq. (2) based on laboratory, regular wave data. The results are consistent with the envelope curve of Fig.
_-
FC
-C
Fig. 2. Top down waves ~ reef data.
251
R. Nelson / Coastal Engineering 32 (1997) 247-254
I developed from regular wave data, and confirm that similar limiting values of H/h apply to the individual waves of natural, irregular, top down, stable wave trains propagating in water of constant depth. It is also important to note that these data sets contained shallow water waves as well as a full range of transitional water waves.
4. Natural,
irregular,
bottom up, stable waves in water of constant depth
All the previous summarized work is now extended and compared with what will be referred to as the bottom up waue case, where natural, random wave trains are generated by wind in water of constant depth (horizontal bed). No shoaling is involved, with the waves growing gradually with distance along the fetch until stationary, fully arisen, sea state conditions are achieved.
0
F
OI'OO
TO'
IO2
IO3
I IO4
deep
OIOO
transitional
10;
Fc
IO3
lo4
Fc
0
-
IO2
shallow
0
deep
transitional
shallow
G
deep
transitional
shallow
I- * d r\l
6
0 OIOO
Fc
IO'
IO2
Fc Fig. 3. Bottom up waves - lake data.
IO3
104
252
R. Nelson/
Coastal Engineering 32 (1997) 247-254
The author had access to field data collected from a large, shallow, inland lake (approximately 10 km X 20 km X 2 m> and collected by Young and Verhagen (1996) while investigating the generation of waves in water of finite depth. The data subset selected for analysis, come from a region where wind waves generated on the relatively shallow, constant depth fetch, achieved fully arisen sea state conditions in high winds, with the sea state stationary with respect to both time and down wind spatial extent. A major limitation to the growth of these waves was the limited water depth, resulting in the extraction of wave energy by bed friction effects. In that sense they were depth limited waves, However, they were not strictly non-breaking waves, but they were waves that were always at the highest achievable limit. Wave height growth was also limited by the maximum wave steepness that could be supported by the wave lengths contained in the fully arisen sea state condition. Any excess energy input over and above that which could be extracted by bed friction, was dissipated by the turbulence of wave breaking, maintaining the constant upper limit of wave height growth. Wave by wave analyses, similar to those made of the top down data, have been made of 4 wave samples collected during the highest wind speed events. The results are shown in Fig. 3 where all waves in each of the 4 wave samples are plotted. Also shown is the envelope curve of Eq. (2) based on laboratory, regular wave data. In Fig. 3, TP is the peak wave energy period, H,,,,, is the spectral estimate of significant wave height, and U is the mean sustained wind speed at a height of 10 m above the water surface. While there are several data points which marginally exceed the limits described by Eq. (2), the bottom LL~ data indicates that the individual waves of natural, irregular, bottom up, stable wave trains, propagatin, 0 in water of constant depth, conform to the same upper H/h limits as top down waves, that limit being the same function (Eq. (2)) developed from regular, laboratory waves. None of the bottom up waves achieved the asymptotic H/h value of 0.55, a limit achieved in both the top down reef experiments and the laboratory experiments. This is probably because none of the bottom up waves even approached shallow water wave conditions. The greatest F, value achieved by any given wave was 200 compared with a required minimum value of 500 for a shallow water wave. In fact, very few waves achieved an F, value of 150. Despite the small water depth and very high wind speed, the wave conditions were at best transitional, with the greatest H/h ratio being 0.50. It seems doubtful whether bottom up, shallow water waves can ever be achieved in water of constant depth. The analyses of the bottom up data, considered in conjunction with the dimensionless functions developed by Young and Verhagen (1996) for the generation of spectral wave parameters in water of finite depth, indicate that the generation of a Oottom up, fully arisen, shallow water sea state, in water of constant depth, is a condition of little engineering significance. Either the water depth needs to be so shallow that it becomes insignificant, or the wind speed so great as to be unrealistic. Even if the wind speeds required were a possibility, it is unlikely that waves as we know them would exist as the water surface would be blown apart. The evidence indicates that truly depth limited, shallow water waves in water of constant depth, can only be achieved by shoaling waves from deeper water into the constant water depth region. What can be concluded with reasonable certainty is that for all bottom up, constant water depth situations of engineering significance, the maximum H/h ratio achievable
R. Nelson/Coastal
Engineering 32 (1997) 247-254
253
does not exceed 0.55, the same upper limit value obtainable for both the top down condition and laboratory wave generation. For completeness, reference is made to Masse1 (1996) who presented the results of a theoretical study on the height limits of bottom up waves in the range of F, < 30. This is much less than the maximum F, values of 200 observed experimentally in this paper. Nevertheless, in the range F, < 30, the results are similar to, but not the same as, the experimental results.
5. Conclusions (1) In water of constant depth, the individual waves in naturally occurring bottom up wave trains, conform to the same upper height limit criteria as those contained in naturally occurring top down wave trains, the limit being defined by the same function developed from monochromatic laboratory wave data, namely Eq. (2) in this paper. (2) In all three wave environments mentioned in 1 above, no individual wave can have a H/h ratio exceeding 0.55, but this upper limit value may be less for bottom up generated waves. (3) In water of constant depth, top down generated wave trains can contain shallow water waves as well as a full range of transitional water waves. However, the field data indicates that bottom up, shallow water waves cannot be achieved in water of constant depth. The best that can be achieved is a partial range of transitional water waves. The maximum achievable F, value is probably about 200 compared with a value of at least 500 required for a shallow water wave. (4) The bottom up generation of shallow water waves in water of constant depth, has no engineering relevance. Either the depth has to be so shallow as to be insignificant, or the wind speed so large as to be unrealistic.
Acknowledgements The author thanks Professor I.R. Young, University lake data.
College, ADFA, for access to the
References CERC (Coastal Engineering Research Center), 1984. Shore Protection Manual, U.S. Army Corps of Engineers, Washington, DC. Coda, Y., 1964. Wave forces on a vertical circular cylinder: experiments and a proposed method of wave force computation, Report No. 8, Port and Harbor Research Institute, Ministry of Transport, Japan. Keating, T., Webber, N.B., 1977. The generation of laboratory waves in a laboratory channel, A comparison between theory and experiment, Proc. Inst. Civil Engineers Part 2, Vol. 63, pp. 819-832. Le Mehaute, B., Divoky, D., Lin, A., 1968. Shallow water waves: a comparison of theories and experiments, Proc. 1lth Coastal Eng. Conf., Vol. 1, London, ASCE, New York, pp. 86-107 Massel, S.R., 1996. Ocean surface waves: their physics and prediction, World Scientific, Advanced Series on Ocean Engineering, Vol. 11.
254
R. Nelson/Coastal
Engineering 32 (19Y7) 247-254
McGowan, J., 1894. On the highest wave of permanent type. Philos. Mag. 38 (5), 351-358. Nelson, R.C., 1982. The effect of bed slope on wave characteristics, Proc. 18th Coastal Eng. Conf., Vol. I, Cape Town, AXE, New York, pp. 555-572. Nelson, R.C., 1985. Wave heights in depth limited conditions. Civil Eng. Trans., Inst. Eng. Aust. 27, 210-215. Nelson, R.C., 1987. Design wave heights on very mild slope?. Civil Eng. Trans., Inst. Eng. Aust. 29, 157-161. Nelson, R.C., 1994. Depth limited design wave heights in very flat regions. Coastal Eng. 23, 43-59. Swart, D.H., Loubser, C.C., 1979. Vocoidal wave theory: Vol. 2: verification, Research Report 360, NRIO, CSIR, South Africa. Young, I.R., Verhagen, L.A., 1996. The growth of fetch limited waves in water of finite depth. Part I: Total energy and peak frequency. Coastal Eng. 29, 47-78.
Coastal Engineering
32 (1997)
Technical
247-254
Note
Height limits in top down and bottom up wave environments Ray Nelson School qf Cil’il Engineering,
*
lJniuer& of New South Wales, Unirxr-sity Collrgr, Academy, Canberra, ACT 2600, Australia
Amtmliarr
llpfcrlce Fnrcc
Keceived 23 January 19Y7; accepted 20 June 1997
Abstract This paper isolates two environments or processes in which maximum stable wave heights can be attained in water of constant depth. In fop down environments the maximum stable wave height, compatible with the depth, is approached from above, while in bottom up bottom up wave trains are shown to conform to the same upper height limit criteria as naturally occurring top down wave trains, that limit being the same function developed from monochromatic laboratory waves. However, important differences are found in the shallow and transitional wave content of the wave populations generated by the two different processes. 0 1997 Elsevier Science B.V. Keywords: Waves;
Constant depth; Depth limited wave heights; Design wave heights
1. Introduction Design height
wave
that can
heights
in neat-shore
be sustained
regions
are often
in the prevailing
water
governed depths,
by the maximum
establishing
wave
a requirement
for depth limited, design wave height criteria. Most project locations are associated with finite bed slopes, but there will always be a significant number of sites located in constant water depth regions (horizontal bed). In Australia, for example, fixed and floating structures are often located on the flat tops of coral reef platforms. Navigation
* Tel.:
t 61-Z-62688330;
FAX:
+61-Z-62688337;
0378.383Y/97/$17,00 0 1997 Elsevier PII s0378-3839(97)00024-0
E-mail:
[email protected].
Science B.V. All rights reserved
R. Nelson/Coastal
248
Engineering
32 (1997) 247-254
aids have been constructed on offshore sand banks in water depths of up to 8 m where the bed level has varied by less than 300 mm over an area of 4 km2. A submerged ocean pipeline has been designed for a cyclone prone area in a constant water depth location so extensive that a water depth of 15 m is persistent over a 10 km length of the pipeline. Nelson, 1982, 1985, 1987, 1994 has shown that the depth limited wave heights that can be sustained in water of constant depth are much lower than those that can be sustained even on very mild slopes (say = 0.01 and even less). The subject therefore has very important design implications. This paper reviews the previous laboratory and field work done by the author on this topic. The work is then extended by identifying two natural processes that produce stable wave trains in water of constant depth, introduces new field data, and compares the depth limited nature of the waves produced by the two processes. lmportant similarities and differences are found. The two wave environments producing the two processes have been called top down and bottom up wave environments. In top down environments, the maximum stable wave height compatible with the existing constant water depth, is approached from above. For example, waves shoaled into the constant depth region, at heights not sustainable by the water depth, break, dissipate energy, and reduce in height until the maximum stable wave height is achieved. High waves propagated from deep water into the constant water depth on a coral reef platform is an example of a top clown wave environment. Bottom up waves approach the maximum stable wave height, compatible with the existing water depth, from below. The waves are at their lowest height when they first exist in the constant water depth region, and must grow in height to achieve the maximum stable wave height. Depth limited wind waves generated on a large shallow lake of uniform depth is an example of a bottom up wave environment.
2. Regular wave data in water of constant depth In a series of papers, Nelson, 1982, 1985, 1987, 1994 developed a function from laboratory data, defining the maximum possible value of H/h for stable oscillatory waves, propagating in water of constant depth. All the data were from monochromatic (regular) wave experiments. The data and the derived function are shown in Fig. 1 (CERC, 1984; Goda, 1964; Keating and Webber, 1977; Le Mehaute et al., 1968; Nelson, 1985, 1994) where use has been made of a non-linearity parameter (F,) after Swart and Loubser (1979). This parameter is a function of T, h and H as given in Eq. (1) where T is the wave period, h is the water depth, and H is the wave height. F, = ‘
g1.25H0.50
T2.50
,$I .I5
This is a particularly useful parameter because it is a function of variables that may be directly measured at a point, even over a sloping bed, a property not achievable if wave length (L) was involved (e.g., Ursell’s parameter). The main implication is that waves of equal F, have approximately the same relative wave shape. Approximate values of F,
R. Nelson/Coastal
Engineering
32 (1997) 247-254
249
corresponding to the classical classification of shallow, transitional, and deep water waves are; shallow water waves F, > 500 transitional water waves 10 < F, < 500 deep water waves F, < 10 Some of the data shown in Fig. 1 are from data sets in which the waves are known to have been at or near the limit of stability (wave breaking). These data are labelled according to the data sets from which they were extracted. For data sets where no distinction was made between waves at or near the limit of stability and those well below that limit, all the data have been plotted under the common label others. The envelope curve defining the maximum H/h ratio is described by Eq. (2). H -= h
F,
(2)
22 + 1.82F,
In shallow water, this function is asymptotic to H/h = 0.55, a value at odds with common engineering practice and the limiting H/h ratio for horizontal beds given by CERC (1984) shown in Fig. 1. This assumed a limiting H/h value of 0.78 based on the work of McGowan (1894) using solitary waves (not oscillatory waves). The heavy dotted line in Fig. 1 represents the deep water steepness limit of 0.142 (Nelson, 1994). It is known that some of the waves shown in Fig. 1 were shoaled into the region of constant water depth, while others are from experiments where a continuous horizontal floor existed between the wave generator and the wave observation station. Fig. 1 includes two data sets of limiting H/h ratios for regular waves over a bed slope of 0.01 (Goda, 1964; Nelson, 1987). They demonstrate how bed slopes as small as this influence the wave mechanics sufficiently to increase the limiting value of H/h to 0.8 and greater.
deep
/
transitiohol
shallow
co
Nelson
(1985),
slope
0.000,
data
set
1
Nelson
(1985),
slope
0.000,
data
set
2
Le
d
Mehaute
Keating
and
Others,
slope
Goda
(1964),
et
01
(1968).
Webber
(1977),
slope
Nelson
(1987),
Nelson
(1985.1994),
slope
0.010 0.010 slope
“1W
CERC
(1984),
slope
0.000
CERC
(1984).
slope
0.010
H/L=O. 142
1o4 Fc Fig. 1. Monochromatic,
laboratory
wave data.
slope
0.000
(H,‘h)max=Fc/(22+1.82Fc)
_-_-_-
slope
0.000
0.000 0.000
250
R. Nelson/Coastal
Engineering
32 (1997) 247-254
Nelson (1987) undertook laboratory experiments to determine how the limiting ratios for H/h vary on bed slopes between horizontal and 0.01. Using six slopes, it was determined that Eq. (3) described the upper limit of H/h for stable, unbroken, oscillatory waves in shallow water, where p is the bed slope angle. = 0.55 + 0.88exp(
3. Natural,
irregular,
-O.O12cot(
/3))
(3)
top down, stable waves in water of constant depth
Nelson (1994) extended the above work to the analysis of field data of wave trains propagating over a broad, horizontal, coral reef platform. 72 data sets were analyzed which covered a wide range of incident wave height and reef top water depth combinations. In about 75% of these data sets, the wave trains were shoaled by the reef front to the extent that they contained breaking waves at the start of the constant water depth region. Wave energy was then dissipated by the turbulence associated with the wave breaking process, until stable wave heights, compatible with reef top water depths, were attained. The experimental observations were made at a point where the stable, oscillatory waves had reformed. These waves are referred to as top down waues and the data will be referred to as top down duta. Wave by wave analyses of these data give rise to the results shown in Fig. 2. These take into account any effects tide or infragravity wave activity may have had in varying the effective water depth on which the wind waves were superimposed. The left side of Fig. 2 shows only one wave from each data set, namely that corresponding to the greatest value of H/h observed in each data set. The right side of Fig. 2 shows all waves from all data sets. Also shown is the envelope curve of Eq. (2) based on laboratory, regular wave data. The results are consistent with the envelope curve of Fig.
_-
FC
-C
Fig. 2. Top down waves ~ reef data.
251
R. Nelson / Coastal Engineering 32 (1997) 247-254
I developed from regular wave data, and confirm that similar limiting values of H/h apply to the individual waves of natural, irregular, top down, stable wave trains propagating in water of constant depth. It is also important to note that these data sets contained shallow water waves as well as a full range of transitional water waves.
4. Natural,
irregular,
bottom up, stable waves in water of constant depth
All the previous summarized work is now extended and compared with what will be referred to as the bottom up waue case, where natural, random wave trains are generated by wind in water of constant depth (horizontal bed). No shoaling is involved, with the waves growing gradually with distance along the fetch until stationary, fully arisen, sea state conditions are achieved.
0
F
OI'OO
TO'
IO2
IO3
I IO4
deep
OIOO
transitional
10;
Fc
IO3
lo4
Fc
0
-
IO2
shallow
0
deep
transitional
shallow
G
deep
transitional
shallow
I- * d r\l
6
0 OIOO
Fc
IO'
IO2
Fc Fig. 3. Bottom up waves - lake data.
IO3
104
252
R. Nelson/
Coastal Engineering 32 (1997) 247-254
The author had access to field data collected from a large, shallow, inland lake (approximately 10 km X 20 km X 2 m> and collected by Young and Verhagen (1996) while investigating the generation of waves in water of finite depth. The data subset selected for analysis, come from a region where wind waves generated on the relatively shallow, constant depth fetch, achieved fully arisen sea state conditions in high winds, with the sea state stationary with respect to both time and down wind spatial extent. A major limitation to the growth of these waves was the limited water depth, resulting in the extraction of wave energy by bed friction effects. In that sense they were depth limited waves, However, they were not strictly non-breaking waves, but they were waves that were always at the highest achievable limit. Wave height growth was also limited by the maximum wave steepness that could be supported by the wave lengths contained in the fully arisen sea state condition. Any excess energy input over and above that which could be extracted by bed friction, was dissipated by the turbulence of wave breaking, maintaining the constant upper limit of wave height growth. Wave by wave analyses, similar to those made of the top down data, have been made of 4 wave samples collected during the highest wind speed events. The results are shown in Fig. 3 where all waves in each of the 4 wave samples are plotted. Also shown is the envelope curve of Eq. (2) based on laboratory, regular wave data. In Fig. 3, TP is the peak wave energy period, H,,,,, is the spectral estimate of significant wave height, and U is the mean sustained wind speed at a height of 10 m above the water surface. While there are several data points which marginally exceed the limits described by Eq. (2), the bottom LL~ data indicates that the individual waves of natural, irregular, bottom up, stable wave trains, propagatin, 0 in water of constant depth, conform to the same upper H/h limits as top down waves, that limit being the same function (Eq. (2)) developed from regular, laboratory waves. None of the bottom up waves achieved the asymptotic H/h value of 0.55, a limit achieved in both the top down reef experiments and the laboratory experiments. This is probably because none of the bottom up waves even approached shallow water wave conditions. The greatest F, value achieved by any given wave was 200 compared with a required minimum value of 500 for a shallow water wave. In fact, very few waves achieved an F, value of 150. Despite the small water depth and very high wind speed, the wave conditions were at best transitional, with the greatest H/h ratio being 0.50. It seems doubtful whether bottom up, shallow water waves can ever be achieved in water of constant depth. The analyses of the bottom up data, considered in conjunction with the dimensionless functions developed by Young and Verhagen (1996) for the generation of spectral wave parameters in water of finite depth, indicate that the generation of a Oottom up, fully arisen, shallow water sea state, in water of constant depth, is a condition of little engineering significance. Either the water depth needs to be so shallow that it becomes insignificant, or the wind speed so great as to be unrealistic. Even if the wind speeds required were a possibility, it is unlikely that waves as we know them would exist as the water surface would be blown apart. The evidence indicates that truly depth limited, shallow water waves in water of constant depth, can only be achieved by shoaling waves from deeper water into the constant water depth region. What can be concluded with reasonable certainty is that for all bottom up, constant water depth situations of engineering significance, the maximum H/h ratio achievable
R. Nelson/Coastal
Engineering 32 (1997) 247-254
253
does not exceed 0.55, the same upper limit value obtainable for both the top down condition and laboratory wave generation. For completeness, reference is made to Masse1 (1996) who presented the results of a theoretical study on the height limits of bottom up waves in the range of F, < 30. This is much less than the maximum F, values of 200 observed experimentally in this paper. Nevertheless, in the range F, < 30, the results are similar to, but not the same as, the experimental results.
5. Conclusions (1) In water of constant depth, the individual waves in naturally occurring bottom up wave trains, conform to the same upper height limit criteria as those contained in naturally occurring top down wave trains, the limit being defined by the same function developed from monochromatic laboratory wave data, namely Eq. (2) in this paper. (2) In all three wave environments mentioned in 1 above, no individual wave can have a H/h ratio exceeding 0.55, but this upper limit value may be less for bottom up generated waves. (3) In water of constant depth, top down generated wave trains can contain shallow water waves as well as a full range of transitional water waves. However, the field data indicates that bottom up, shallow water waves cannot be achieved in water of constant depth. The best that can be achieved is a partial range of transitional water waves. The maximum achievable F, value is probably about 200 compared with a value of at least 500 required for a shallow water wave. (4) The bottom up generation of shallow water waves in water of constant depth, has no engineering relevance. Either the depth has to be so shallow as to be insignificant, or the wind speed so large as to be unrealistic.
Acknowledgements The author thanks Professor I.R. Young, University lake data.
College, ADFA, for access to the
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